David Cameron is facing growing pressure at home and abroad for the UK to take in more of the many thousands of people fleeing to Europe.
It comes after a picture of a dead boy lying on a Turkish beach sparked an outcryover the crisis’s human cost.
Ex-Tory party chairman Baroness Warsi said the UK should be prepared to “share the burden” and a senior UN official said the UK could “do more”.
On Wednesday, the prime minister said taking more people was not the answer.
Meanwhile, an online petition calling on the UK to accept more refugees has passed the 100,000 threshold, meaning it is eligible to be considered for a debate in Parliament.
It says the UK “is not offering proportional asylum” in comparison with other EU countries.
A number of Conservative MPs have called for the government to take in more migrants, with backbencher David Burrowes saying the UK “should accept thousands, not hundreds” of people.
Johnny Mercer, the Conservative MP for Plymouth Moor View, said that mothers trying to keep their children afloat on life jackets should not think of the UK as a place that did not welcome them.
And Stafford MP Jeremy Lefroy said he believed the UK should provide sanctuary to more refugees, alongside ongoing efforts to tackle the causes of the crisis.
Analysis, BBC political editor Laura Kuenssberg
Only the hardest of hearts would not be moved by the latest images of the people who have been caught up in the danger and chaos of this mass movement.
Images of a toddler, washed up on a beach in Turkey, ram home the sometimes deadly consequences of what is going on.
Right now though there seems little prospect of the UK government shifting its position on taking in more refugees, as EU leaders and Labour opponents have been calling for.
Although these latest images are heart wrenching, David Cameron’s conviction is that it’s more effective to tackle the cause not the symptom of this crisis.
However, a Downing Street source suggested to the BBC that calls for the UK to accept more refugees would not be heeded.
The source said that since last year Britain had accepted 216 people under a scheme to relocate the most vulnerable refugees, and almost 5,000 Syrians had been granted asylum in the last four years.
‘Passing the buck’
Baroness Warsi, a former Foreign Office minister, said that while Britain had been “incredibly generous” in its support for Syrian refugees it could do more in response to the “unfolding crisis” in Europe.
She told BBC Radio 4’s Today programme Britain had a “long and proud” tradition of helping in times of crisis, and Britain should say to the rest of the Europe “that we will, too, share the burden”.
Lady Warsi suggested the government should consider accepting more women and children refugees “genuinely” fleeing war and persecution – but that a “firm” line should be taken against economic migrants.
“This is not about having an open door policy. This is about having quite a specific responsive policy in the areas for example that we have expertise,” she told the BBC, drawing comparisons with the situation in the 1930s when thousands of unaccompanied children fleeing persecution in Europe were allowed in to the UK.
Labour leadership candidate Andy Burnham is urging Mr Cameron to call an emergency debate and vote on the issue when Parliament returns from its summer recess next week.
“The government needs to change its policy today and present its plan to Parliament,” he told Sky News, asking “where have they been all summer as this tragedy has been unfolding across Europe?”.
He added: “This is everybody’s problem. To start passing the buck in the way that our government are doing, the Hungarian government are doing, I think it misses the point entirely.”
Fellow candidate Yvette Cooper has written to Mr Cameron asking him to take in more refugees, and described the current situation as the “greatest humanitarian crisis to reach our continent since the Second World War”.
Peter Sutherland, the United Nations special representative on international migration, said while some countries were “massively bearing the burden”, the UK was among those that “can do more”.
He told BBC Newsnight: “The only way to solve this problem is by a united European response and that means sharing responsibility for appalling suffering.
“This is a humanitarian crisis that Europe has not experienced in our time, of a dimension which demands a common response.”
He said that Germany, Sweden, France and Italy were all taking in more refugees, proportionate to their size, than the UK.
But former Conservative International Development Secretary Andrew Mitchell told the programme that the UK was providing an “immense amount of humanitarian support” to countries such as Syria.
“Were Britain not providing that support, there would be yet more hundreds of thousands of people coming out of that part of the world,” he said.
He added that the UK had done “more than the whole of the EU put together in terms of financial support”.
Meanwhile, Hungary’s Prime Minister, Viktor Orban, is to meet European leaders in Brussels for talks later about the ongoing migrant crisis. Hundreds of migrants have been stranded at a railway station in Hungary’s capital, Budapest.
They were prevented from getting on trains to leave Hungary, but the main railway station in Budapest has now opened its doors to those who have been gathered there for the past two days.